Generalized exponents of primitive symmetric digraphs (Q1356508)

The exponent of a non-bipartite graph is defined to be the minimum integer \(t\) such that given any pair of vertices \(x\) and \(y\) there is a walk from \(x\) to \(y\) of length \(t\) (and hence of all lengths \(p\geq t\)). Let \(\gamma(x)\) be the smallest integer \(q\) such that for each vertex \(y\) of a graph, there is a walk of length \(q\) from \(x\) to \(y\). Then the vertices can be ordered \(v_1,\dots, v_n\) so that \(\gamma(v_1)\leq\dots \leq\gamma(v_n)\) (so that \(\gamma(v_n)\) is the exponent). \(\gamma(v_k)\) is said to be the \(k\)th generalized exponent of the graph. In this paper the largest value of values of these generalized exponents is determined, over all graphs whose shortest odd cycle has length (a fixed number) \(r\). The extremal graphs are found also. Note that these exponents are more commonly considered for digraphs.